Understanding of buoyancy in drill pipe and risers

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Faculty of Science and Technology

MASTER’S THESIS

Study program/ Specialization:

Offshore technology – Marine and Subsea Technology

Spring semester, 2013

Open / Restricted access Writer:

Morten Reve ………

(Writer’s signature)

Faculty supervisor: Prof. Arnfinn Nergaard External supervisor(s):

Title of thesis:

Understanding of Buoyancy in Drill Pipe and Risers

Credits (ECTS): 30 Key words:

Buoyancy, Marine Riser, Drill Pipe, Effective Tension, Bridgman

Pages: 78

Stavanger, 17.06.2013 Date/year

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Preface

This thesis is written as a final closure for my master degree program in Offshore Technology; Subsea Technology. The thesis was conducted from January to June in 2013, at the University of Stavanger.

The thesis is about how buoyancy is interpreted in general and when calculating top tension in risers and drill pipes.

I would like to thank my supervisor Arnfinn Nergaard for his patience during this thesis. Many hours have been spent discussing the topic of this thesis. I would also like to thank professors Rune W.

Time, Bernt S. Aadnoy and Bjørn H. Hjertager for their contributions during this thesis.

A special thanks goes to technician Arne Haaverstein from IRIS for his help making the pressure vessel needed to conduct the Bridgman experiment.

Last but not least I would like to thank my girlfriend Malene Skogly for her patience, support and understanding during this semester.

This thesis has helped me understand how buoyancy and hydrostatic forces contributes when calculating the needed top tension in risers and drill pipes.

Morten Reve

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Summary

This thesis highlights that there is a wide spread use of the term “buoyancy” in the petroleum industry which can lead to misunderstandings. It is also evident that terms like “effective tension”,

“true wall tension” and “apparent weight” induces more misunderstandings if not understood correctly.

It has been shown that there is different ways of interpreting buoyancy forces and how they act on a submerged object. Several experiments have been used as illustrations to show that fluid need non- vertical sides to create a lift force (buoyancy force) on an object immersed in fluid although the object is displacing fluid.

Further on it has been shown that the effective tension concept used in marine riser calculations can be misinterpreted because of the different buoyancy understandings. When calculating the effective tension the influence of the horizontal pressure acting on the riser is accounted for. In other terms, the effective tension is a three dimensional stress calculation which gives the needed top tension force (in one dimension) to prevent buckling as an answer. This calculation can be interpreted as if there were a “buoyancy force” present along the entire length of the riser, which is a contradiction to what has been presented in this thesis.

An experiment has been conducted to show that effective tension concept gives correct results of the internal three dimensional stress state and thus is not just a “fictitious stress” or “fictitious force”

as mentioned in several papers, but a stress state which can lead to failures if not accounted for.

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List of tables and figures Tables

Table 6.1: Calculated rupture pressure for the PTFE rod ... 48

Table 6.2: Calculated rupture pressure for the PMMA rod ... 49

Table 6.3: Pressure readings and geometries of PTFE rods before and after test ... 50

Table 6.4:Pressure readings at rupture of PMMA rods ... 50

Table A.1: Numerical values ... 65

Table A.2: Calculated friction due to compression ... 65

Table A.3: Ar value... 66

Figures

Figure 2.1: Generalized steps of a drilling operartion [7] ... 5

Figure 2.2: Drilling risers with buoyancy modules to the left - 3D figure of the riser, drill string and kill/choke/auxiliary lines to the right [9][10] ... 7

Figure 2.3: Main components of a marine riser [8] ... 9

Figure 3.1: Archimedes' Law [1] ... 13

Figure 3.2: Pressure and weight acting in a fluid [1] ... 13

Figure 3.3: Archimedes Law by superposition [1] ... 14

Figure 3.4: Internal forces acting on a submerged body segment [1] ... 15

Figure 3.5: Pipe with internal fluid - equivalent force system [1] ... 16

Figure 3.6: Pipe with internal and external fluids - equivalent force systems [1] ... 17

Figure 4.1: Body immersed in fluid, showing only the vertical forces from the fluid ... 22

Figure 4.2: Experiment showing Archimedes’ Principle [24] ... 23

Figure 4.3: Experiment setup [25] ... 24

Figure 4.5: Step 1 ... 26

Figure 4.9: Experiment setup ... 28

Figure 5.1: Solid bar standing on bottom with no fluid in between ... 31

Figure 5.2: Solid bar hanging in vacuum and internal force distribution ... 32

Figure 5.3 Internal forces acting in a solid bar immersed in fluid interpreted by the Archimedes School ... 32

Figure 5.4: Internal forces acting in a solid bar immersed in fluid interpreted by the Piston Force School ... 33

Figure 5.5: Comparison of both schools interpretation of internal forces ... 34

Figure 5.6: Solid bar standing on the bottom with no fluid ... 35

Figure 5.7: Illustration used in combination with equation (5.1) [36] ... 36

Figure 6.1: Setup for Bridgman experiment ... 38

Figure 6.2: Bridgman's explanations of the "Pinching Off" effect. [42] ... 39

Figure 6.3: High pressure unit ... 40

Figure 6.4: High pressure unit; External air and water intake and high pressure water outtake ... 41

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Figure 6.5: Pressure vessel (Bridgman’s chamber) connected to high pressure hose ... 41

Figure 6.6: PTFE rod after test, a close look will show the initiated yielding and reduction of the outer diameter ... 42

Figure 6.7: PMMA rod prior to testing, the cone in the end helps during introducing the rod through the seal rings ... 42

Figure 6.8: High pressure unit connected to the pressure vessel ... 43

Figure 6.9: Pressure vessel and ruptured rod parts inside a wooden crate... 44

Figure 6.10: PTFE Stress vs. Strain in tension [44] ... 45

Figure 6.11: PMMA stress vs. strain in tension [46] ... 46

Figure 6.12: Friction forces from the sealing rings ... 47

Figure 6.13: Static friction force as function of pressure ... 48

Figure 7.1: Effective tension in the bar derived by Hubbert and Rubey ... 52

Figure 7.2: Interpretation of the effective tension ... 52

Figure 7.3: Nergaard’s illustration of effective tension [35] ... 53

Figure 7.4: Stress components at a point in loaded body [39] ... 53

Figure 7.5: Tensile strength test ... 54

Figure 7.6: Pressure loading as in the Bridgman experiment ... 55

Figure 7.7: Elongation in y-direction [48] ... 56

Figure 7.8: Elongation in x-direction [48] ... 56

Figure 7.9: Stress in x-direction [48] ... 56

Figure 7.10: Stress in y-direction [48] ... 56

Figure 7.11: Shear stress [48] ... 57

Figure 7.12: Von Mises stress [48] ... 57

Figure 8.1: Solid bar standing on bottom ... 58

Figure 8.2: Solid bar standing on bottom. Effective tension equal to 0 at lower end ... 59

Figure 8.3: Interpretation of Bridgman experiment using effective tension ... 59

Figure A.1: Fluid pressure squeezes the sealing ring and increases the normal force between the rod and sealing ring [47] ... 63

Figure A.2: Showing the different dimensions needed to do the calculations [47] ... 64

Figure A.3: Diagrams used to calculate the friction force [47] ... 65

Figure A.4: Estimated fh curve ... 66

Figure A.5: Calculated Fh as function of pressure ... 66

Figure A.6: Calculated dynamic friction force ... 67

Figure A.7: Calculated static friction ... 67

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Chapter 1. Introduction

This thesis’ purpose is to investigate how the term “buoyancy” is interpreted from a volume

perspective and a pressure perspective. These two interpretations are then used to show that there is a need for clarification of the term when dealing with top tension calculations on marine risers and drill pipes.

An experiment will be conducted; the results from the experiment will be used to show that effective tension can be used to calculate the internal stress state of an object. We will also create an analogy on how the effective tension concept, a concept used to calculate the top tension needed to prevent a riser string to buckle, can be interpreted.

The thesis will be divided into 8 chapters. Chapter 1 contains the introduction, objectives and the background for the thesis. In chapter 2 we will have an introduction to the oil and gas industry in general and a more specific introduction to marine riser systems and components of such systems.

Chapter 3 will introduce us to the effective tension concept as derived by C. P. Sparks. In chapter 4 we take a closer look at how text books in physics introduce students to buoyancy, discuss the two views and in the end define the term “buoyancy”. Chapter 5 will introduce two schools of

understanding buoyancy in the petroleum industry which is based on the discussion in chapter 4. A literature study on the subject has been performed. Chapter 6 will be devoted to an experiment conducted to show how effective tension describes a “real” force. In chapter 7 there will be a discussion based on the results from the experiment. Chapter 8 will contain a conclusion and a recommendation for further work.

1.1 Objectives

- Give an introduction to the offshore petroleum industry and marine riser systems

- Introduce the effective tension concept – a concept used to calculate the top tension needed to prevent a marine riser from buckling

- Investigate how buoyancy is understood in basic physics and illustrate that there is a need for clarification of the term “buoyancy” when and object is placed at the bottom of a fluid container

- Based on the buoyancy investigation and a literature study, show that misinterpretation of the term “buoyancy” when doing calculations using the effective tension concept can occur - Show that effective tension is not a fictitious force by conducting an experiment. Discuss the

results from the experiment

- Prepare conclusions and recommendations for further work

1.2 Background

When doing calculations on top tension needed to prevent marine risers from buckling the effective tension concept is often used. [1] There seems to be much confusion on what effective tension means, and how to derive the equations used. To further complicate things other terms like

“effective axial force”, “true wall tension”, “apparent weight”, “buoyancy method”, “pressure area method” are introduced. “True wall tension” can again be referred to as “true force”, “true tension”,

“real tension”, “actual tension” or “absolute tension” and effective tension can sometimes be termed

“fictive tension” or “absolute tension less end cap load”. It has even been called “the effective weight of the effective mass”. [1][2]

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2 C. P. Sparks is the author often referred to when effective tension is mentioned in offshore

standards. [1][3] In his book [1] he states that “a suspended riser will see a buoyancy force equal to the weight of the fluid displaced, which for a vertical riser of uniform cross section is equal to the pressure x area (𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒) acting at the riser lower end. Note, however, that the buoyancy force acts at the centroid of the submerged volume, at the midheight of the submerged length, not at the riser lower end”. [1] In this thesis we will review this statement and the effective tension concept and present how we can understand effective tension.

In practice a marine riser will never be perfectly vertical or stand firmly on the sea bottom. In fact it is very hard to perform measurements on a real riser determining the tension forces which are acting, as there is as so many different parameters in play (currents, waves, rig motion).

During this thesis we will only be looking at perfectly vertical bars of uniform cross section. We will only investigate the pressure area term (𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒) in the effective tension equation presented by Sparks, thus the bars will be solid with no internal volume. This will exclude any form of stress induced by drilling mud inside the riser. We will also exclude any auxiliary lines. All cases are static cases.

In a newer book [4] the effective tension is described as; ”The effective force is not a true force in that it cannot be measured with a strain gauge or weight indicator.” Further on it is described as “The effective force is a fictitious quantity …”. In this thesis we will try to show that by calculating the effective force one can predict failures. [4]

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Chapter 2. Introduction to the oil and gas industry

Parts of chapter 2 have been used in an earlier project at the University of Stavanger. [5]

2.1 Offshore development

Oil has been used for over five thousand years. In the middle east oil seeped up from the ground.

This oil was used for waterproofing boats and baskets, paint, provide light and even in medication.

[6]

Whale oil has been used as source of light in recent times. But because of the increased use of whale oil the whale population dropped and this increased an already high oil price further. [6]

The demand for oil was then much higher than the supply and many companies and individuals were looking for a larger and more lasting source of the what-to-be-known-as “black gold”. The answer came with the development of drilling for crude oil onshore. The demand for oil did still rise and this led to the exploration companies to look for oil below the seabed. [6]

Prior to the Second World War drilling offshore was limited to shallow waters of Lake Maracibo, Venezuela and the swamps and coastal area of Louisiana in the US. After the Second World War there was a significant change in the oil industry as America was making its transition from a war- time to a peace-time economy. Until then the government had controlled the oil-price, but now the states started to dispute over the offshore shelf mineral rights. There was a large public demand for oil and gas, and the companies encountered challenges e.g. underwater exploration, weather forecasting, tidal and current prediction, drilling location determination and offshore

communications. [6]

Despite these challenges, the first well was drilled from a fixed platform offshore out-of-sight of land in 1947. The combination of a barge and platform was a significant breakthrough in drilling-unit design for offshore use. This event marked the beginning of the modern offshore industry as it is known today. [6]

In the Gulf of Mexico, the first oil well structures to be built in open waters were in the water depths of up to 100m and based on a piled jacket structure, in which a framed template has piles driven through it to pin the structure to the sea bed. To this, a support frame was added for the working parts of the rig, such as deck and accommodation. These structures were the fore-runners for the massive platforms in many locations around the world, including the North Sea. [6]

There was a high activity level in the oil industry in the 1960s. Many new offshore oil and gas fields were discovered in the Gulf of Mexico, the Southern North Sea, the South China Sea and in the Gulf of Suez. [6]

Two “oil shocks” in the 1970s led to dramatic increases in oil prices and a perception that oil was in short supply. [6]

In the 1970s and the early 1980s there were years of unprecedented offshore activity. The high oil prices and a perceived need to increase security of oil supplies facilitated the installation of giant platforms in the hostile waters of the Northern North Sea and offshore Alaska, and in the Gulf of Mexico. [6]

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4 In 1986 a collapse in the oil price put the future of such offshore field developments into question. As a response the industry came up with innovative solutions that enabled new developments to go ahead. The oil price was low for almost twenty years, but after 2004 there has been a significant increase in both oil and gas prices. There has also been a significant increase of costs for exploration, development and operation after 2004. Safety and environmental issues has also been a growing concern which the industry has been facing. [6]

The industry has in recent times been looking towards smaller fields, often with complex geology, and remote and frontier areas, for example deep water. This is because it is not likely that there will be discovered many new “giant” offshore fields although large volumes of oil and gas lies offshore.

To develop these resources economically, the industry has to find new solutions that combine cost- effectiveness over the lifetime of the project with improved safety and environmental performance.

[6]

2.2 Current activities and trends

The offshore oil and gas industry has for the last few decades developed very high activity in the North Sea, the Gulf of Mexico, the South China Sea, offshore Brazil and offshore West Africa. The North Sea has been the largest producing region of the offshore oil industry. There are a few deep water fields being developed offshore Norway and West of Shetland, but most reservoirs are in water less than 200 meters deep. Development of satellite fields is a major feature in this region – generally small accumulations of oil and gas which lie close to existing production facilities. [6]

The current focus of the industry includes the continental shelf of Gulf of Mexico, offshore Brazil and West Africa (e.g. Nigeria and Angola) where water depths reach some 3000 meters. Many of the record breaking developments in deep offshore drilling have been in these locations. [6]

“Frontier” areas such as offshore Alaska and the Barents Sea are also in the focus of the industry.

Remoteness, deep water, high winds, floating ice and sub-zero temperatures are some of the challenges the industry faces in these areas. Some arctic regions are frozen up to 10 months of the year, putting severe limitations on the drilling activities. [6]

Western companies have increased their interest in the former USSR’s hydrocarbon resources after the opening of the country. More than a fifth of the world’s offshore oil and gas resources could be located here according to some estimates. To date only a small part of the area has been explored. In the Sea of Okhotsk north of Japan and in the Barents and Kara Seas in the Russian Arctic there have been a number of oil and gas field discoveries. [6]

2.3 Deepwater development challenges and trends

Expanding into deepwater frontier areas with no existing infrastructure has always been a challenge.

[6]

Many achievements, which are comparable with the space industry, have lead to the possible development of the offshore oil industry in hostile waters. Many fields are located far from land and operations are extended to even more remote locations. New fields are being explored in ever deeper and harsher waters, like the Norwegian Sea, the Atlantic Ocean west of Scotland and the Barents Sea. [6]

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2.4 Drilling

A marine drilling riser system is used during a drilling operation offshore. The drilling process can differ from operation to operation but the basic steps in each operation can be summarized as shown in figure 2.1:

Figure 2.1: Generalized steps of a drilling operartion [7]

- Step 1: Lower a temporary guide base which is connected to 4 guide wires. This will help guide different equipment in the following steps.

- Step 2: Start drilling through the seabed with a 30” or 36” drill bit to around 120m. This drilling sequence is done without a riser. Use a rope to guide the drill pipe in place as shown in figure 2.2

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6 - Step 3: After the drilling in step 2 is completed the 30” conductor will be lowered into the

hole and cemented. The upper part of the conductor is the wellhead housing. A permanent guide base is installed.

- Step 4: A 26” hole is drilled to approximately 500m.

- Step 5: The 20” casing with wellhead is lowered and cemented in place.

- Step 6: The subsea Blow-out Preventer (BOP), Lower Marine Riser Package (LMRP) and the marine riser is installed and the drilling continues. Drill mud will now circulate through the marine riser bringing the drill cuttings up to the platform where it is removed and the mud is pumped down the well again. The BOP, LMRP and marine riser will be connected to the wellhead for the rest of the drilling operation. [6][8]

2.5 Marine riser system

Marine risers were first used to drill from barges offshore California in the 1950s. In 1961 an important landmark occurred when drilling took place from the dynamically positioned (DP) barge CUSS-1. Since those early days, risers have been used for four main purposes [1]:

- Drilling

- Completion/workover - Production/injection - Export

In each of the four groups of risers there is a large variety of details, dimensions and materials. In this thesis we will focus only on the drilling riser. Drilling risers can be divided into low-pressure and high- pressure risers. Their difference is discussed in section 2.6.1 and 2.6.2. [1]

2.5.1 Low-pressure drilling riser

A low-pressure riser is open to atmospheric pressure in the top end. This is the standard drilling riser used today. Because it is open in the top end the internal pressure can never be higher than that owing to the drilling-mud weight. Drilling risers are made of up joints which can vary between 15-23 meters (typically) in length. The nominal diameter of the central tube is usually 21” and on the outside the central tube is equipped with several auxiliary lines, see figure 2.2. If the BOP is closed due to a kick in the well the kill and choke lines are used to communicate with the well and to circulate fluid. The booster line is used to inject fluids in the lower end of the riser to accelerate the drill cutting flow back to the surface. The two small auxiliary lines shown in figure 2.2 can be hydraulic lines, which is used to power the subsea BOP. [1]

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Figure 2.2: Drilling risers with buoyancy modules to the left - 3D figure of the riser, drill string and kill/choke/auxiliary lines to the right [9][10]

Riser joints can be fitted with syntactic foam buoyancy modules to increase the buoyancy force which reduces the weight of the joint in water, see figure 2.2. The upper part of a drilling riser is usually fitted with modules with the exception of the splash zone. This to reduce the hydrodynamic forces the waves will induce on the riser in this area. The design pressure for the modules increases with the water depth implying a stronger, heavier and denser design. This favors installing buoyancy modules at the top end of the riser. [1]

Air-can buoyancy modules have been used in the past. The advantage with these was that they could be optimized for each individual drilling campaign. The disadvantage was that they added a level of complexity. [1]

The connector in the top and the bottom of the riser joint is another feature which can have many different designs. [1]

The auxiliary lines are supported by guide clamps. The design of the guide clamps is critical because they prevent the lines from buckling. Another good practice which ensures that the safety level is kept is to design the lines so they cannot break out of their housing at the connector level, even if the line should buckle. [1]

2.5.2 High-pressure drilling riser

A high-pressure drilling riser is used when the BOP is located at the surface, as was the case for the CUSS-1 in 1961. In the event of a kick the BOP is accessible for closing from the drill rig and no subsea choke and kill lines are necessary. Because of the lack of choke and kill lines along the submerged riser the architecture is much simpler than a low-pressure riser. The riser must be designed to take the full well pressure. However, when drilling with a surface BOP there Is potentially more risk, unless an adequate seal and disconnect system can be provided in case of an emergency. [1]

High pressure risers with surface BOPs have been used to drill from many tension leg platforms since the 1980s, such as Hutton (1984), Heidrun, Mars, RamPowell, and URSA, and from some spars. In the case of the Heidrun TLP, the drilling riser was made of titanium. [1]

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8 In the early 1990s a high-pressure slimline (small-diameter) riser with surface BOP was proposed for the Ocean Drilling Program, to allow scientific drilling with mud circulation in ultra deep water (>4000 m). But the project was not pursued. High-pressure risers with surface BOPs have more recently been used to drill a large number of wells from semisubmersibles in moderate environmental conditions.

The concept continues to be developed for deeper water and harsher environments. [1]

2.6 Components of a Marine Drilling Riser System

The marine drilling riser system is a continuation of the well bore from the seabed to the surface. It connects the subsea BOP Stack to the drilling vessel. [11]

According to the American Petroleum Institute (API) specification 16F [11] the main function of the marine riser system is to:

- Provide for fluid communication between the drilling vessel and the BOP Stack and the well:

o Through the main bore during drilling operations

o Through the choke and kill lines when the BOP Stack is being used to control the well o Through the auxiliary lines such as hydraulic fluid supply and mud boost lines

- Guide tools into the well

- Serve as a running and retrieving string for the BOP Stack The main components of a marine riser are shown in figure 2.3.

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Figure 2.3: Main components of a marine riser [8]

2.6.1 Upper Marine Riser Package (UMRP)

The upper part of the riser string including the riser tensioner system is called the UMRP. [12] The UMRP includes:

- The diverter system - Upper flex joint

- Self-tensioned slip joint (telescopic joint) and tensioner ring - Riser rotation bearing joint

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10 2.6.1.1 Diverter

The diverter diverts the drill mud and cuttings from going vertically up from the riser system and potentially be blown out on the drill floor in case of a kick, routing the fluid horizontally out and into flowlines connected to the mud system. [13]The surface diverter is mounted on top of the UMRP but it is not part of the marine riser system. [12]

2.6.1.2 Upper Flex/ball joint

The upper flex joint or ball joint is positioned on surface level in between the diverter and the inner barrel of the slip joint. The joint allow some misalignment of angle between the riser system and the drilling vessel (roll pitch and offset motions of the vessel). [12]

2.6.1.3 Telescopic joint and tensioner ring

The telescopic joint, or the slip joint as it is often named, consists of an outer and inner barrel. The outer barrel is attached to the riser string and is held in tension by wire ropes or hydraulic cylinders from the top end of the outer barrel to the tensioners. [13] The inner barrel is connected to the upper flex/ball joint and can move freely inside the outer barrel to compensate for the drilling vessels horizontal and vertical movement. Between the outer and inner barrel there is a packer element which seals of the annulus, this prevents fluid leakage from the riser. On the top end of the outer barrel there is typically mounted or incorporated a tensioner ring. The main function of the tensioner ring is to transmit the support load from the riser tensioner lines to the outer barrel. Some tensioner ring systems also allow for rotation of the vessel around the riser. The telescopic joint usually have terminal fittings for connecting the choke, kill and auxiliary line drape hoses to the rigid lines used on the riser joints. [12]

2.6.1.4 Riser Rotation Bearing Joint

The riser rotation bearing joint is mounted at the bottom of the telescopic joint. It allows the drilling vessel to rotate around the riser’s vertical axis and minimizes the torque transferred from the riser to the telescopic joint. It typically consists of a roller bearing system, built in locking device and

hydraulic motors. The hydraulic motors and the built in locking device is used for precise rotational control and preventing inappropriate rotation of the riser. [12]

2.6.2 Riser joints See chapter 2.5.1

2.6.3 Lower Marine Riser Package (LMRP)

The lower marine riser package is an assembly located at the bottom of the drilling riser, but above the BOP. The LMRP provides releasable interface between the riser and BOP stack. [12]

Typical component in a LMRP are: [11]

- Lower Riser Adapter - Flex/ball joint bypass lines - Lower flex/ball joint

- Hydraulic connectors for mating the riser to the BOP stack 2.6.3.1 Lower Riser Adapter

The lower riser adapter is the connection between the lower most riser joint and the lower flex/ball joint mounted on the lower marine riser package. [11]

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11 2.6.3.2 Flex/Ball joint bypass lines

The bypass lines are mounted on kick outs on the riser adapter. They bypass the flex/ball joint and terminate in the BOP. [11]

2.6.3.3 Lower flex/ball joint

See section 2.7.1.2 as upper and lower flex joint is basically the same. [11]

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Chapter 3. Effective Tension Concept

To be able to present the effective tension concept correct, section 3.1 is solely based on chapter 2 and chapter 3 from the book “Fundamentals of Marine Riser Mechanics” [1] written by C. P. Sparks.

In section 3.2 we will discuss the concept based on the assumptions mentioned in section 1.2.

3.1 Effective Tension

“All codes of practice require the global behavior of pipes and risers to be calculated using effective tension. This is generally defined in one of two ways: [1]

- Some codes quote equation (3.1)

- Some codes mention that effective tension is the axial tension calculated at any point of the riser by considering only the top tension and the apparent weight of the intervening riser segment.”

𝑇𝑇_𝑒𝑒 =𝑇𝑇_{𝑡𝑡𝑡𝑡} − 𝑝𝑝_𝑖𝑖𝐴𝐴_𝑖𝑖+𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒 (3.1) Where

𝑇𝑇_𝑒𝑒 =𝐸𝐸𝐸𝐸𝐸𝐸𝑒𝑒𝐸𝐸𝑡𝑡𝑖𝑖𝐸𝐸𝑒𝑒 𝑡𝑡𝑒𝑒𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡

𝑇𝑇𝑡𝑡𝑡𝑡 =𝑇𝑇𝑇𝑇𝑇𝑇𝑒𝑒 𝑡𝑡𝑤𝑤𝑤𝑤𝑤𝑤 𝑡𝑡𝑒𝑒𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑖𝑖𝑡𝑡 𝑤𝑤𝑖𝑖𝑇𝑇 𝑤𝑤𝑡𝑡 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑡𝑡𝐸𝐸 𝑖𝑖𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡 𝑝𝑝_𝑖𝑖 =𝑖𝑖𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝐸𝐸𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓 𝑝𝑝𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇𝑒𝑒 𝑤𝑤𝑡𝑡 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑡𝑡𝐸𝐸 𝑖𝑖𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡 𝐴𝐴_𝑖𝑖 =𝑖𝑖𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝐸𝐸𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑒𝑒𝐸𝐸𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤 𝑤𝑤𝑇𝑇𝑒𝑒𝑤𝑤

𝑝𝑝_𝑒𝑒 =𝐸𝐸𝐸𝐸𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝐸𝐸𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓 𝑝𝑝𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇𝑒𝑒 𝑤𝑤𝑡𝑡 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑡𝑡𝐸𝐸 𝑖𝑖𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡 𝐴𝐴_𝑒𝑒 =𝐸𝐸𝐸𝐸𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝐸𝐸𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑒𝑒𝐸𝐸𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤 𝑤𝑤𝑇𝑇𝑒𝑒𝑤𝑤

To calculate needed top tension in a riser system to prevent buckling Sparks introduces “The Effective Tension Concept”. This concept includes the influence of the tension in the riser walls, internal and external pressure and the weight of the pipe. [1]

3.1.1 Archimedes’ Law

“Archimedes’ Law states in its most general form states that when a body is wholly or partially immersed in a fluid, it experiences an upthrust equal to the weight of fluid displaced. This is illustrated in figure 3.1, in which a body is shown fully immersed in a fluid”. [1]

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13

Figure 3.1: Archimedes' Law [1]

The argument taught to school children is that the pressure field is just able to maintain the displaced fluid in equilibrium, as shown in figure 3.2. Thus, it must provide an upthrust 𝑈𝑈 equal to the weight of the fluid displaced 𝑊𝑊_𝐸𝐸. Furthermore, since this upthrust can produce no rotation, it must act at the centroid of the displaced fluid, which is also the center of gravity 𝐺𝐺. Hence, it will also act at the centroid of the submerged body. [1]

Figure 3.2: Pressure and weight acting in a fluid [1]

Thus, if the true weight of the body is 𝑊𝑊_𝑡𝑡, the tension in the string will be given by the following, where 𝑊𝑊_𝑡𝑡− 𝑊𝑊_𝐸𝐸 is generally called the apparent weight 𝑊𝑊_𝑤𝑤: [1]

𝑇𝑇=𝑊𝑊_𝑡𝑡− 𝑈𝑈=𝑊𝑊_𝑡𝑡− 𝑊𝑊_𝐸𝐸 (3.2)

There are a number of important points to make about Archimedes’ Law: [1]

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14 - The law can be applied directly only to pressure fields that are completely closed. Note that

for a suspended or floating body, the pressure field appears not to be closed; however, since the pressure at the surface is zero, the field can be considered to be closed.

- The law cannot be applied directly to parts of submerged bodies, such as that below the dotted line in figure 3.1.

- The law says nothing about internal forces or stresses.

- The closed pressure field, when combined with the distributed weight of the displaced fluid, can produce no resultant moment. The fluid would not be able to support the associated stresses.”

3.1.2 Archimedes’ Law by superposition

Archimedes’ Law can also be deduced by superposition. This may be too abstract for school children, but leads to the same results more clearly and directly. Since superposition will be used extensively in this book, it will be first used here to rederive Archimedes’ Law. [1]

In figure 3.3, the two systems shown (the submerged body and the displaced fluid) are both in equilibrium under the combine loads that include the effects of tension, pressure, and weight. Hence, if the two systems are superimposed and the forces on the displaced fluid are subtracted from those on the submerged body, the resulting equivalent system will also be in equilibrium. [1]

Figure 3.3: Archimedes Law by superposition [1]

Superposition of the two systems allows the identical pressure fields to be eliminated. All that remains in the resulting equivalent system is the tension 𝑇𝑇 in the string and the apparent weight 𝑊𝑊𝑤𝑤, which is then simply the difference between the weights of the submerged body and the displaced fluid, as given by equation (3.3). [1]

𝑊𝑊_𝑤𝑤 =𝑊𝑊_𝑡𝑡− 𝑊𝑊_𝐸𝐸 (3.3)

Any two systems can be superimposed in this way. The only requirement is that they both be in equilibrium. In the preceding, there is no need to specify that densities must be constant or that the upthrust acts at the centroid of one or other of two systems. The argument can be applied directly to cases where the submerged body does not have a constant density; where the body is suspended

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15 across the interface between fluids of different densities, or where the density of the displaced fluid may vary vertically according to some law. As long as the displaced fluid segment represents exactly the fluid displaced by the submerged body, superposition can be used directly. [1]

3.1.3 Internal forces in a submerged body

In the calculation of the internal forces on a part of a submerged body, the problem is to take into account the pressure filed that is not closed. Figure 3.4 shows the forces acting on the segment below the dotted line in figures 3.1 and 3.3. The resultant of the pressure filed acting on the underside of the segment is unknown and cannot be determined directly using Archimedes Law. [1]

Figure 3.4: Internal forces acting on a submerged body segment [1]

Nevertheless, superposition allows the internal forces to be determined very simply. The middle sketch of figure 3.4 shows the forces acting on the displaced fluid segment including the closed pressure field. If these forces are subtracted from the forces on the body segment, the pressure field acting below the body is conveniently eliminated. However, the force 𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒, owing to the pressure acting on the section, remains (where 𝑝𝑝𝑒𝑒 is the pressure in the fluid and 𝐴𝐴𝑒𝑒 is the cross-sectional area of the section). Since convention requires tension to be positive, this must be shown as a tensile force:

−𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒. [1]

The equivalent system (figure 3.4, right-hand sketch) shows the resultant of the superposition. Once again, the apparent weight 𝑊𝑊_𝑤𝑤 is given by equation (3.3), where the weights 𝑊𝑊_𝑡𝑡, 𝑊𝑊_𝐸𝐸 and 𝑊𝑊_𝑤𝑤

correspond to the segment, rather than the whole body. Thus, the apparent weight 𝑊𝑊𝑤𝑤 is in equilibrium with an effective tension 𝑇𝑇𝑒𝑒, a shear force 𝐹𝐹, and a moment 𝑀𝑀, which can be found by resolving forces normal and parallel to the section and by taking moments. The shear force 𝐹𝐹 and the moment 𝑀𝑀 are the same as on the body segment. (For the applications considered in this book, the minute moment created by the very slight pressure gradient across the section can be neglected.) The effective tension 𝑇𝑇𝑒𝑒 is then related to the true tension 𝑇𝑇𝑡𝑡𝑇𝑇𝑇𝑇𝑒𝑒 as shown in equation (3.4). [1]

𝑇𝑇_𝑒𝑒 =𝑇𝑇_{𝑡𝑡𝑇𝑇𝑇𝑇𝑒𝑒} −(−𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒) =𝑇𝑇_{𝑡𝑡𝑇𝑇𝑇𝑇𝑒𝑒} +𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒 (3.4)

According to convention, tensile forces are positive. However, according to a further convention, pressures are also positive. The positive sign in the right-hand side of equation (3.4) results from the

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16 contradiction between the two conventions. The effective tension 𝑇𝑇_𝑒𝑒 is nevertheless the difference between the tensions acting on the body segment and the displaced fluid segment, just as apparent weight is the difference between their weights”. [1]

3.1.4 Curvature, Deflections and Stability of Pipes and risers under pressure

“The preceding arguments can be extended to the case of pipes and risers under pressure. Figure 3.5 shows equivalent force systems for the case of a pipe subjected to internal pressure 𝑝𝑝_𝑖𝑖. For clarity, moments and shear forces have been omitted, but that does not influence the argument. A pipe segment of length 𝛿𝛿𝑡𝑡 is shown curved and in equilibrium under the combined influence of pipe weight, internal pressure, and the true wall tension 𝑇𝑇𝑡𝑡𝑡𝑡 acting on the pipe wall. [1]

Figure 3.5: Pipe with internal fluid - equivalent force system [1]

The pressure field acting on the internal fluid column is closed and in equilibrium with the weight of the internal fluid. The lateral pressures acting on the pipe wall are equal and opposite of those acting on the internal fluid. Hence, by superposition and addition of the two force systems, those lateral pressures are eliminated. However, the axial “tension” in the fluid column −𝑝𝑝_𝑖𝑖𝐴𝐴_𝑖𝑖 remains (where 𝑝𝑝_𝑖𝑖 is the internal pressure and 𝐴𝐴𝑖𝑖 is the internal cross-sectional area of the pipe). This leads to the

equations for the effective tension 𝑇𝑇_𝑒𝑒 and apparent weight 𝑡𝑡_𝑤𝑤 of the equivalent system: [1]

𝑇𝑇_𝑒𝑒 =𝑇𝑇_{𝑡𝑡𝑡𝑡} + (−𝑝𝑝_𝑖𝑖𝐴𝐴_𝑖𝑖) (3.5) 𝑡𝑡_𝑤𝑤 =𝑡𝑡_𝑡𝑡+𝑡𝑡_𝑖𝑖 (3.6)

When external pressure 𝑝𝑝_𝑒𝑒 is also present, the same approach can still be used, as shown in figure 3.6. By the addition of the force systems acting on the pipe segment and the internal fluid and then the subtraction of the force system acting on the displaced fluid, all lateral pressure effects are

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17 eliminated. In figure 3.6, 𝑡𝑡_𝑡𝑡, 𝑡𝑡_𝑖𝑖, 𝑡𝑡_𝑒𝑒 and 𝑡𝑡_𝑤𝑤 are the weights per unit length of the tube, the internal fluid column, the displaced fluid column and the equivalent system, respectively. [1]

Figure 3.6: Pipe with internal and external fluids - equivalent force systems [1]

The equations for the effective tension 𝑇𝑇_𝑒𝑒 and the apparent weight 𝑡𝑡_𝑤𝑤 then becomes 𝑇𝑇𝑒𝑒 =𝑇𝑇𝑡𝑡𝑡𝑡 + (−𝑝𝑝𝑖𝑖𝐴𝐴𝑖𝑖)−(−𝑝𝑝𝑒𝑒𝐴𝐴𝑒𝑒) (3.7)

𝑡𝑡𝑤𝑤 =𝑡𝑡𝑡𝑡+𝑡𝑡𝑖𝑖− 𝑡𝑡𝑒𝑒 (3.8)

Furthermore, the two concepts are related, as can be seen from the right-hand sketch in figure 3.6.

For an element of length 𝛿𝛿𝑡𝑡, resolution of forces in the axial direction gives

𝑓𝑓𝑇𝑇_𝑒𝑒

𝑓𝑓𝑡𝑡 =𝑡𝑡_𝑤𝑤𝐸𝐸𝑡𝑡𝑡𝑡 𝛹𝛹 (3.9)

which for small angles with the vertical becomes ^{𝑓𝑓𝑇𝑇}^𝑒𝑒

𝑓𝑓𝑡𝑡 =^{𝑓𝑓𝑇𝑇}_{𝑓𝑓𝐸𝐸}^𝑒𝑒=𝑡𝑡_𝑤𝑤. [1]

Since for any fluid the combined effects of its weight and enclosed pressure field can produce no resultant moment anywhere, the bending effects of forces on the equivalent system are precisely the same as those on the pipe segment. Therefore, the simplest way to take into account the effects of internal and external pressure on pipe or riser curvature, deflection and stability is to use effective tension and apparent weight in the corresponding tensioned-beam calculations. [1]

The effective tension, at any point along a riser, can be obtained most simply by considering the equilibrium of the segment between the point and the riser top end, taking into account the riser top tension and the segment apparent weight. The true wall tension 𝑇𝑇_{𝑡𝑡𝑡𝑡} can then be found from equation (3.7)”. [1]

3.1.5 Confusion regarding buoyancy

In chapter 3, “Application of Effective Tension – Frequent Difficulties and Particular Cases”, of Sparks’

book [1], he again discusses buoyancy;

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18

“The Archimedes upthrust, or buoyancy, acting on a submerged body was recalled at the beginning of chapter 2. It is a volumetric force in the sense that it is the resultant of the closed pressure field acting on the enclosed volume. It is equal to the weight of fluid displaced by the body and acts at the

centroid of the submerged volume. The concept can be applied to any fully submerged body. It can also be applied to any suspended or floating body – such as a suspended riser or a ship hull – for which the pressure fields can be considered to be closed. For a suspended riser, the buoyancy is equal to the weight of fluid displaced, which for a vertical riser of uniform section is equal to the pressure x area (𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒) acting at the riser lower end. Note, however, that the buoyancy force acts at the centroid of the submerged volume, at the midheight of the submerged length, not at the riser lower end. Ships would capsize if buoyancy acted at the keel level instead of at the centroid of their displaced volume.

[1]

Confusion arises when discussing the buoyancy of part of a submerged object (see figure3.4), such as a segment of riser, since it is subject to a pressure field that is not closed. The confusion is particularly flagrant if the riser pipe concerned is vertical and of uniform section. The wall of such a riser is continuous. Hence, the fluid pressure will act only horizontally on the segment and will have no vertical component. It is tempting to say that such a riser segment has no buoyancy. Since the

segment can be positioned anywhere along the riser length, that would imply that the entire riser has no buoyancy, except at the surface at the lower end. That plainly does not agree with Archimedes’

conception of buoyancy as an upthrust acting on a submerged volume. [1]

The confusion is further increased when considering the stability of a vertical uniform riser connected to the sea bed, since the external fluid pressures the do not even apply a vertical force to the surface at the riser lower end! Yet if the riser has negative apparent weight (i.e., is lighter than water), it will remain vertical and stable even if the top tension is reduced to zero. How does it do that without collapsing in a heap on the seabed? Some would argue that if the riser did depart from vertical, forces with vertical components would be generated which would return it to the vertical”. [1]

3.2 Effective tension - Discussion

Equation (3.7) and (3.8) derived by Sparks will in this thesis be reduced to equation (3.11) and (3.12) since the annulus and drilling mud is excluded due to the assumptions made in section 1.2.

𝑇𝑇_𝑒𝑒 =𝑇𝑇_{𝑡𝑡𝑡𝑡} −(−𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒) (3.11) 𝑡𝑡_𝑤𝑤 =𝑡𝑡_𝑡𝑡− 𝑡𝑡_𝑒𝑒 (3.12)

The required top tension 𝑇𝑇 to prevent the riser from buckling at any point (lower end in practice), considering the whole length of the riser and the effective tension equal to zero at the bottom of the pipe, will then become

𝑇𝑇=𝑡𝑡𝑤𝑤 =𝑡𝑡𝑡𝑡− 𝑡𝑡𝑒𝑒 =𝑡𝑡𝑡𝑡− 𝑝𝑝𝑒𝑒𝐴𝐴𝑒𝑒 (3.12) Where

𝑇𝑇=𝑀𝑀𝑖𝑖𝑡𝑡𝑖𝑖𝑀𝑀𝑇𝑇𝑀𝑀 𝑡𝑡𝑡𝑡𝑝𝑝 𝑡𝑡𝑒𝑒𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑝𝑝𝑇𝑇𝑒𝑒𝐸𝐸𝑒𝑒𝑡𝑡𝑡𝑡 𝑏𝑏𝑇𝑇𝐸𝐸𝑏𝑏𝑤𝑤𝑖𝑖𝑡𝑡𝑏𝑏 𝑡𝑡_𝑤𝑤 =𝑤𝑤𝑝𝑝𝑝𝑝𝑤𝑤𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡 𝑡𝑡𝑒𝑒𝑖𝑖𝑏𝑏ℎ𝑡𝑡 𝑡𝑡𝐸𝐸 𝑇𝑇𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇

𝑡𝑡_𝑡𝑡 =𝑡𝑡𝑒𝑒𝑖𝑖𝑏𝑏𝑡𝑡ℎ 𝑡𝑡𝐸𝐸 𝑇𝑇𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇 𝑖𝑖𝑡𝑡 𝑤𝑤𝑖𝑖𝑇𝑇

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19 𝑡𝑡_𝑒𝑒 =𝑡𝑡𝑒𝑒𝑖𝑖𝑏𝑏ℎ𝑡𝑡 𝑡𝑡𝐸𝐸 𝑓𝑓𝑖𝑖𝑡𝑡𝑝𝑝𝑤𝑤𝑤𝑤𝐸𝐸𝑒𝑒𝑓𝑓 𝐸𝐸𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓

𝑝𝑝_𝑒𝑒 =𝑒𝑒𝐸𝐸𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝑝𝑝𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇𝑒𝑒 𝑤𝑤𝑡𝑡 𝑤𝑤𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇 𝑒𝑒𝑡𝑡𝑓𝑓 𝑡𝑡𝐸𝐸 𝑇𝑇𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇 𝐴𝐴_𝑒𝑒 =𝑒𝑒𝐸𝐸𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝐸𝐸𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑒𝑒𝐸𝐸𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑤𝑤𝑡𝑡 𝑤𝑤𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇 𝑒𝑒𝑡𝑡𝑓𝑓 𝑡𝑡𝐸𝐸 𝑇𝑇𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇

Comparing equation (3.12) with equation (3.13) given by API 16 Q [14] to calculate the minimum slip ring tension

𝑇𝑇_{𝑆𝑆𝑆𝑆𝑀𝑀𝑖𝑖𝑡𝑡} =𝑊𝑊_𝑡𝑡𝐸𝐸_{𝑡𝑡𝑡𝑡} − 𝐵𝐵_𝑡𝑡𝐸𝐸_{𝑏𝑏𝑡𝑡} +𝐴𝐴_𝑖𝑖(𝑓𝑓_𝑀𝑀𝐻𝐻_𝑀𝑀 − 𝑓𝑓_𝑡𝑡𝐻𝐻_𝑡𝑡) (3.13) Where

𝑇𝑇_{𝑆𝑆𝑆𝑆𝑀𝑀𝑖𝑖𝑡𝑡} =𝑀𝑀𝑖𝑖𝑡𝑡𝑖𝑖𝑀𝑀𝑇𝑇𝑀𝑀 𝑆𝑆𝑤𝑤𝑖𝑖𝑝𝑝 𝑆𝑆𝑖𝑖𝑡𝑡𝑏𝑏 𝑇𝑇𝑒𝑒𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 (𝑡𝑡𝑡𝑡 𝑤𝑤𝐸𝐸𝑡𝑡𝑖𝑖𝑓𝑓 𝑏𝑏𝑇𝑇𝐸𝐸𝑏𝑏𝑤𝑤𝑖𝑖𝑡𝑡𝑏𝑏 𝑤𝑤𝑡𝑡 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡) 𝑊𝑊𝑡𝑡 =𝑆𝑆𝑇𝑇𝑏𝑏𝑀𝑀𝑒𝑒𝑇𝑇𝑏𝑏𝑒𝑒𝑓𝑓 𝑆𝑆𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇 𝑊𝑊𝑒𝑒𝑖𝑖𝑏𝑏ℎ𝑡𝑡 𝑤𝑤𝑏𝑏𝑡𝑡𝐸𝐸𝑒𝑒 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑓𝑓𝑒𝑒𝑇𝑇𝑒𝑒𝑓𝑓 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡

𝐸𝐸_{𝑡𝑡𝑡𝑡} =𝑆𝑆𝑇𝑇𝑏𝑏𝑀𝑀𝑒𝑒𝑇𝑇𝑏𝑏𝑒𝑒𝑓𝑓 𝑊𝑊𝑒𝑒𝑖𝑖𝑏𝑏ℎ𝑡𝑡 𝑇𝑇𝑡𝑡𝑤𝑤𝑒𝑒𝑇𝑇𝑤𝑤𝑡𝑡𝐸𝐸𝑒𝑒 𝐹𝐹𝑤𝑤𝐸𝐸𝑡𝑡𝑡𝑡𝑇𝑇 (𝑀𝑀𝑖𝑖𝑡𝑡1,05) 𝐵𝐵_𝑡𝑡 =𝑁𝑁𝑒𝑒𝑡𝑡 𝐿𝐿𝑖𝑖𝐸𝐸𝑡𝑡 𝑡𝑡𝐸𝐸 𝐵𝐵𝑇𝑇𝑡𝑡𝐵𝐵𝑤𝑤𝑡𝑡𝐸𝐸𝐵𝐵 𝑀𝑀𝑤𝑤𝑡𝑡𝑒𝑒𝑇𝑇𝑖𝑖𝑤𝑤𝑤𝑤 𝑤𝑤𝑏𝑏𝑡𝑡𝐸𝐸𝑒𝑒 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑓𝑓𝑒𝑒𝑇𝑇𝑒𝑒𝑓𝑓 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝐸𝐸_{𝑏𝑏𝑡𝑡} =𝐵𝐵𝑇𝑇𝑡𝑡𝐵𝐵𝑤𝑤𝑡𝑡𝐸𝐸𝐵𝐵 𝑤𝑤𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑡𝑡𝑓𝑓 𝑇𝑇𝑡𝑡𝑤𝑤𝑒𝑒𝑇𝑇𝑤𝑤𝑡𝑡𝐸𝐸𝑒𝑒 𝐹𝐹𝑤𝑤𝐸𝐸𝑡𝑡𝑡𝑡𝑇𝑇 (𝑀𝑀𝑤𝑤𝐸𝐸0,96) 𝐴𝐴𝑖𝑖 =𝐼𝐼𝑡𝑡𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝐶𝐶𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑒𝑒𝐸𝐸𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝐴𝐴𝑇𝑇𝑒𝑒𝑤𝑤 𝑡𝑡𝐸𝐸 𝑆𝑆𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇 𝑤𝑤𝑡𝑡𝑓𝑓 𝑤𝑤𝑇𝑇𝐸𝐸.𝑤𝑤𝑖𝑖𝑡𝑡𝑒𝑒𝑡𝑡 𝑓𝑓_𝑀𝑀 =𝐷𝐷𝑇𝑇𝑖𝑖𝑤𝑤𝑤𝑤𝑖𝑖𝑡𝑡𝑏𝑏 𝐹𝐹𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓 𝑊𝑊𝑒𝑒𝑖𝑖𝑏𝑏ℎ𝑡𝑡 𝐷𝐷𝑒𝑒𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡𝐵𝐵

𝐻𝐻_𝑀𝑀 =𝐷𝐷𝑇𝑇𝑖𝑖𝑤𝑤𝑤𝑤𝑖𝑖𝑡𝑡𝑏𝑏 𝐹𝐹𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓 𝐶𝐶𝑡𝑡𝑤𝑤𝑇𝑇𝑀𝑀𝑡𝑡 𝑤𝑤𝑏𝑏𝑡𝑡𝐸𝐸𝑒𝑒 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑓𝑓𝑒𝑒𝑇𝑇𝑒𝑒𝑓𝑓 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝑓𝑓_𝑡𝑡 =𝑆𝑆𝑒𝑒𝑤𝑤 𝑊𝑊𝑤𝑤𝑡𝑡𝑒𝑒𝑇𝑇 𝑊𝑊𝑒𝑒𝑖𝑖𝑏𝑏ℎ𝑡𝑡 𝐷𝐷𝑒𝑒𝑡𝑡𝑡𝑡𝑖𝑖𝑡𝑡𝐵𝐵

𝐻𝐻𝑡𝑡 =𝑆𝑆𝑒𝑒𝑤𝑤 𝑊𝑊𝑤𝑤𝑡𝑡𝑒𝑒𝑇𝑇 𝐶𝐶𝑡𝑡𝑤𝑤𝑇𝑇𝑀𝑀𝑡𝑡 𝑤𝑤𝑏𝑏𝑡𝑡𝐸𝐸𝑒𝑒 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑓𝑓𝑒𝑒𝑇𝑇𝑒𝑒𝑓𝑓 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 Note that 𝑊𝑊_𝑡𝑡 can be written as

𝑊𝑊_𝑡𝑡 =𝑡𝑡_𝑡𝑡−(𝐴𝐴_𝑒𝑒 − 𝐴𝐴_𝑖𝑖)𝑓𝑓_𝑡𝑡𝐻𝐻_𝑡𝑡 (3.14) Where

𝑡𝑡_𝑡𝑡 =𝑆𝑆𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇 𝑊𝑊𝑒𝑒𝑖𝑖𝑏𝑏ℎ𝑡𝑡 𝑖𝑖𝑡𝑡 𝑤𝑤𝑖𝑖𝑇𝑇 𝑤𝑤𝑏𝑏𝑡𝑡𝐸𝐸𝑒𝑒 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑓𝑓𝑒𝑒𝑇𝑇𝑒𝑒𝑓𝑓 𝑝𝑝𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡

𝐴𝐴𝑒𝑒 =𝐸𝐸𝐸𝐸𝑡𝑡𝑒𝑒𝑇𝑇𝑡𝑡𝑤𝑤𝑤𝑤 𝐶𝐶𝑇𝑇𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑒𝑒𝐸𝐸𝑡𝑡𝑖𝑖𝑡𝑡𝑡𝑡 𝐴𝐴𝑇𝑇𝑒𝑒𝑤𝑤 𝑡𝑡𝐸𝐸 𝑆𝑆𝑖𝑖𝑡𝑡𝑒𝑒𝑇𝑇 𝑤𝑤𝑡𝑡𝑓𝑓 𝑤𝑤𝑇𝑇𝐸𝐸.𝑤𝑤𝑖𝑖𝑡𝑡𝑒𝑒𝑡𝑡

And by combining equation (3.13) and (3.14) and neglecting the tolerances and lift from buoyancy material we get

𝑇𝑇_{𝑆𝑆𝑆𝑆𝑀𝑀𝑖𝑖𝑡𝑡} =𝑊𝑊_𝑡𝑡−(𝐴𝐴_𝑒𝑒− 𝐴𝐴_𝑖𝑖)𝑓𝑓_𝑡𝑡𝐻𝐻_𝑡𝑡+𝐴𝐴_𝑖𝑖(𝑓𝑓_𝑀𝑀𝐻𝐻_𝑀𝑀− 𝑓𝑓_𝑡𝑡𝐻𝐻_𝑡𝑡) =𝑊𝑊_𝑡𝑡− 𝐴𝐴_𝑒𝑒𝑓𝑓_𝑡𝑡𝐻𝐻_𝑡𝑡+𝐴𝐴_𝑖𝑖𝑓𝑓_𝑀𝑀𝐻𝐻_𝑀𝑀 (3.15) Equation (3.15) is equal to equation (3.12) if we consider a solid riser with no internal cross section;

𝑇𝑇𝑆𝑆𝑆𝑆𝑀𝑀𝑖𝑖𝑡𝑡 =𝑡𝑡𝑡𝑡− 𝐴𝐴𝑒𝑒𝑓𝑓𝑡𝑡𝐻𝐻𝑡𝑡 =𝑡𝑡𝑡𝑡− 𝑝𝑝𝑒𝑒𝐴𝐴𝑒𝑒 (3.16)

It is easy to show that 𝑝𝑝𝑒𝑒𝐴𝐴𝑒𝑒 at the lowest point of the riser is numerical the same as if we would calculate the buoyancy force.

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20 In chapter 4 we will take a closer look on how buoyancy can be understood and define what a

buoyancy force is. This definition will show that the term 𝑝𝑝_𝑒𝑒𝐴𝐴_𝑒𝑒 in equation (3.16) is not a buoyancy force.

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Chapter 4. Buoyancy

In this chapter we will take a closer look on how buoyancy is introduced in modern day physics. We will then review two different methods to interpret buoyancy by calculating buoyancy as pressures on flats, and by calculating buoyancy as weight of displaced volume. Different experiments trying to show that one school is right and the other school is wrong will be presented. In the end of the chapter we will define how a buoyancy force will be interpreted in the rest of this thesis.

Buoyancy is widely used and understood throughout the world. But as shown in this chapter it also generates a good topic of discussion. First we will look at Archimedes’ Principal describing buoyancy.

4.1 Archimedes’ Principle

Some 2000 years ago the great Archimedes defined how buoyancy works on an object immersed or partly immersed in a fluid. Archimedes did consider several cases, or propositions; this includes a solid which is lighter than the fluid it is immersed in, a solid which is equal in weight as the fluid it is immersed in and finally a solid which is heavier than the fluid it is immersed in. [15]

The propositions Archimedes wrote: [15]

- “Proposition 3: Of solids those which size for size, are of equal weight with a fluid will, if let down into the fluid, be immersed so that they do not project above the surface but do not sink lower”

- “Proposition 4: A solid lighter than a fluid will, if immersed in it, not be completely submerged, but part of it will project above the surface”

- “Proposition 5: Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced”

- “Proposition 6: If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced”

- “Proposition 7: A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced”

All the propositions listed above have been proved and accepted. A mathematical explanation has been derived and follows the same propositions exactly.

Modern text books of physics could name the force mentioned in proposition 6 “buoyancy” [16][17],

“buoyancy force” [18], “buoyant force” [19][20][21][22] or “upthrust” [1][23].

A modern way of describing buoyancy is by using a figure, like figure 4.1, and state that the buoyancy acting on the body partly or fully immersed in fluid is the net sum of the vertical forces (the

horizontal forces are equal and opposite, canceling out each other) acting on the lower side of the body and the vertical forces acting on the top side of the body. The forces are naturally calculated from the pressure acting on the lower and upper part of the body; this is shown in equation (4.1).

𝐵𝐵=∆𝐹𝐹=𝐹𝐹_𝑤𝑤− 𝐹𝐹_𝑇𝑇 =𝐴𝐴 ∗ ∆𝑝𝑝=𝐴𝐴 ∗(𝑝𝑝_𝑤𝑤− 𝑝𝑝_𝑇𝑇) =𝐴𝐴 ∗ ℎ ∗ 𝜌𝜌_{𝐸𝐸𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓} ∗ 𝑏𝑏 (4.1) This is known as the piston-force understanding of buoyancy.

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22

Figure 4.1: Body immersed in fluid, showing only the vertical forces from the fluid

A different way to describe the same buoyancy force is to state that the buoyancy force is equal to the volume of the displaced fluid multiplied with the fluids density. The mathematical expression is shown in equation (4.2).

𝐵𝐵=𝑉𝑉𝑓𝑓𝑖𝑖𝑡𝑡𝑝𝑝𝑤𝑤𝑤𝑤𝐸𝐸𝑒𝑒𝑓𝑓 𝐸𝐸𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓 ∗ 𝜌𝜌𝐸𝐸𝑤𝑤𝑇𝑇𝑖𝑖𝑓𝑓 ∗ 𝑏𝑏 (4.2)

This is known as the volume understanding of buoyancy.

It is simple to show that equation (4.1) is equal to equation (4.2). These two different methods of calculating buoyancy will almost always give the same answer when calculating the buoyancy force acting on a body.

It will also be stated that the buoyancy force acts through the center of buoyancy, which is equal to the center of gravity of the displaced fluid.

In the physic text books investigated in this thesis both the above approaches are used.

[16][17][18][19][20][21][22][23]

4.2 Archimedes’s Principle in practice

To prove Archimedes’ Principle many different experiments can be conducted. One of the simplest but most useful experiments which illustrate the effect of buoyancy is shown in figure 4.2.

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23

Figure 4.2: Experiment showing Archimedes’ Principle [24]

The body hanging in the air has a weight of 7 lb as shown by the left hand sketch of figure 4.2. When the body is lowered into the beaker it will displace water (the water level will raise). The water that the body displaces will be drained into the bowl next to the beaker. When the body is completely immersed in fluid, a volume of water equal to the volume of the body will be in the bowl. As the weight shows in the right-hand sketch of figure 4.2 the body, when immersed in fluid, is only

weighing 4 lb. The figure also shows that the water in the bowl is weighing the same as the “missing”

3 lb, or in other words; the buoyancy force is equal to the weight of the displaced fluid.

But if the object is standing firmly on the ground, with no fluid between the ground and the object, will there be any buoyancy force acting on it?

Using the volume interpretation of Archimedes Principle will give a buoyancy force equal to the displaced fluid. That the object is still displacing fluid is evident. But by using the piston force approach one would get a negative buoyancy force, or a force directed downwards.

In the following of this section we will review a couple of experiments or thought experiments used by recent papers dealing with this problem. We will also show an experiment conducted by Goins in 1980 and an experiment conducted at the University of Stavanger during this master project.

4.2.1 Experiment 1: “Do fluid always push up objects immersed in them?” [25]

Figure 4.3 shows a hands-on experiment which is used to show that the buoyancy force will

“disappear” if there is no fluid underneath the object immersed in fluid. Figure 4.3 a) shows a glass box filled with water. On the bottom of the glass box there is a glass prism whose upper surface is optically polished. Below the surface of the water there is floating a table tennis ball with an attached thin polished glass plate. When the ball is pushed down and makes contact with the prism, as shown in figure 4.3 b), the buoy remains standing although there is an upwards force still acting on the ball.

[25]

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24

Figure 4.3: Experiment setup [25]

The author’s explanation of this is simple, as there is no water underneath the plate attached, the water cannot exert a force directed upwards on the plate. Instead, there is a force from the fluid above the plate, directed downwards (the weight of the fluid column above the plate), keeping the plate attached to the prism. The authors conclude after this experiment that the “widespread definition of buoyancy dealing with the amount of water displaced by the body immersed in a fluid is deficient”. It is convenient for calculating the magnitude of the buoyancy force but can be misleading and give an erroneous result if applied without caution”. [25]

Comments: In figure a) the ball seems to be neutral buoyant, and this could lead one to think that the ball will stay in equilibrium if it is moved down into the water. But the authors do comment in the paper that the mass of the ball can be manipulated by adding or removing water inside of the ball, and thus manipulate how large the buoyancy force will be with regards to the mass of the ball.

It can be discussed how long the ball will be kept attached to the prism in practice, as in the end the ball will float up. This can be used as an observation to confirm that the buoyancy force finally counteracted some additional force that held the ball down and thus the original Archimedes Principle explained by the volume understanding of buoyancy is still valid.